耦合含时滞的相互依存网络的局部自适应异质同步

针对由两个子网络构成的耦合含时滞的相互依存网络,研究其局部自适应异质同步问题.时滞同时存在于两个子网络的内部耦合项和子网络间的一对一相互依赖耦合项中,且网络的耦合关系满足非线性特性和光滑性.基于李雅普诺夫稳定性理论、线性矩阵不等式方法和自适应控制技术,通过对子网络设置合适的控制器,提出了使得相互依存网络的子网络分别同步到异质孤立系统的充分条件.针对小世界网络和无标度网络构成的相互依存网络进行数值模拟,验证了提出理论的正确性和有效性.

Local adaptive heterogeneous synchronization for interdependent networks with delayed coupling

With the development of the networks, the coupling between networks has become increasingly significant. Here, the networks can be described as interdependent networks. An interdependent network can have two different kinds of links, a connectivity link and a dependency link, which are fundamental properties of interdependent networks. During the past several years, interdependent complex network science has attracted a great deal of attention. This is mainly because the rapid increase in computing power has led to an information and communication revolution. Investigating and improving our understanding of interdependent networks will enable us to make the networks (such as infrastructures) we use in daily life more efficient and robust. As a significant collective behavior, synchronization phenomena and processes are common in nature and play a vital role in the interaction between dynamic units. At the same time, the time delay problem is an important issue to be investigated, especially in biological and physical networks. As a matter of fact, time delays exist commonly in the real networks. A signal or influence traveling through a network is often associated with time delay. In this paper, the local adaptive heterogeneous synchronization is investigated for interdependent networks with delayed coupling consisting of two sub-networks, which are one-by-one inter-coupled. The delays exist both in the intra-coupling and in the inter-coupling between two sub-networks, the intra-coupling and inter-coupling relations of the networks satisfy the requirements for nonlinearity and smoothness, and the nodes between two sub-networks have different dynamical systems, namely heterogeneous systems. Based on the Lyapunov stability theory, linear matrix inequality, and adaptive control technique, with proper controllers and adaptive laws for the networks, the sufficient conditions are proposed to synchronize the sub-networks of the interdependent networks into heterogeneous isolated systems, respectively. In order to illustrate the main results of the theoretical analysis clearly, some numerical simulations for an interdependent network with NW small world sub-network and BA sub-network are presented, in which each sub-network has 100 nodes and the heterogeneous systems are Lorenz and Rössler systems. The numerical simulations show that using the controllers and adaptive laws proposed, the network obtains the local heterogeneous synchronization quickly, that is, the nodes of two sub-networks are synchronized into Lorenz and Rössler systems separately. Thus, they verify the feasibility and correctness of the proposed techniques. It is worth noting that the presented results are delay-independent. In the future, our research will be directed to the further investigation of the delay-dependent synchronization of interdependent networks by using the current results as a basis.